Optimal Scaling of a Gradient Method for Distributed Resource Allocation

We consider a class of weighted gradient methods for distributed resource
allocation over a network. Each node of the network is associated with a local
variable and a convex cost function; the sum of the variables (resources)
across the network is fixed. Starting with a feasible allocation, each node
updates its local variable in proportion to the differences between the
marginal costs of itself and its neighbors. We focus on how to choose the
proportional weights on the edges (scaling factors for the gradient method) to
make this distributed algorithm converge, and how to make the convergence as
fast as possible. We give sufficient conditions on the edge weights for the
algorithm to converge monotonically to the optimal solution; these conditions
have the form of a linear matrix inequality. We give some simple, explicit
methods to choose the weights that satisfy these sufficient conditions. We also
derive a guaranteed convergence rate for the algorithm, and find the weights
that minimize this rate by solving a semidefinite program. Finally, we extend
the main results to problems with general equality constraints, and problems
with block separable objective function.